Set \( t \leftarrow t+1\)  and update \(\big (N_a(t)\big )_{a \in [K]}\) .

Empirical set of \(\varepsilon \) -good arms: \(G_\varepsilon (\widehat{{\mu }}_{\tau _\delta })\) 


Now we state our sample complexity result which we adapted from Theorem 14 in [1]}. Notably, while their Track-and-Stop strategy relies on tracking the exact optimal weights to prove that the expected stopping time matches the lower bound when \(\delta \)  tends to zero, our proof shows that it is enough to track some slightly sub-optimal weights with a decreasing gap in the order of \(\frac{1}{\sqrt{t}}\)  to enjoy the same sample complexity guarantees.
